# American Institute of Mathematical Sciences

August  2019, 24(8): 3947-3970. doi: 10.3934/dcdsb.2018338

## Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay

* Corresponding author

Dedicated to Peter E. Kloeden on the occasion of his 70th birthday

Received  June 2018 Revised  August 2018 Published  August 2019 Early access  January 2019

Fund Project: The authors were partly supported by MINECO/FEDER grant MTM2015-66330-P, and the European Commission under project H2020-MSCA-ITN-2014 643073 CRITICS.

This paper provides a dynamical frame to study non-autonomous parabolic partial differential equations with finite delay. Assuming monotonicity of the linearized semiflow, conditions for the existence of a continuous separation of type Ⅱ over a minimal set are given. Then, practical criteria for the uniform or strict persistence of the systems above a minimal set are obtained.

Citation: Rafael Obaya, Ana M. Sanz. Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3947-3970. doi: 10.3934/dcdsb.2018338
##### References:
 [1] R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.  Google Scholar [2] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., 1964.  Google Scholar [3] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, Berlin, Heidelberg, New York, 1981.  Google Scholar [4] R. Johnson, R. Obaya, S. Novo, C. Núñez and R. Fabbri, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control, Developments in Mathematics 36, Springer, Switzerland, 2016. doi: 10.1007/978-3-319-29025-6.  Google Scholar [5] O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moskow, 1967 (Russian). English transl.: Transl. Math. Monographs, AMS, Providence, 1968.  Google Scholar [6] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications Vol. 16, Birkhäuser, Basel, Boston, Berlin, 1995. doi: 10.1007/978-3-0348-9234-6.  Google Scholar [7] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc. 321 (1990), 1-44. doi: 10.2307/2001590.  Google Scholar [8] R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35.   Google Scholar [9] J. Mierczyński and W. Shen, Lyapunov exponents and asymptotic dynamics in random Kolmogorov models, J. Evol. Equ., 4 (2004), 371-390. doi: 10.1007/s00028-004-0160-0.  Google Scholar [10] S. Novo, C. Núñez, R. Obaya and A. M. Sanz, Skew-product semiflows for non-autonomous partial functional differential equations with delay, Discrete Continuous Dynam. Systems - A, 34 (2014), 4291-4321.  doi: 10.3934/dcds.2014.34.4291.  Google Scholar [11] S. Novo, R. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dynamics Differential Equations, 25 (2013), 1201-1231. doi: 10.1007/s10884-013-9337-y.  Google Scholar [12] S. Novo, R. Obaya and A. M. Sanz, Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows, Nonlinearity, 26 (2013), 2409-2440.  doi: 10.1088/0951-7715/26/9/2409.  Google Scholar [13] C. Núñez, R. Obaya and A. M. Sanz, Minimal sets in monotone and sublinear skew-product semiflows Ⅱ: Two-dimensional systems of differential equations, J. Differential Equations, 248 (2010), 1899-1925. doi: 10.1016/j.jde.2009.12.006.  Google Scholar [14] R. Obaya and A. M. Sanz, Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems, J. Differential Equations, 261 (2016), 4135-4163. doi: 10.1016/j.jde.2016.06.019.  Google Scholar [15] R. Obaya and A. M. Sanz, Is uniform persistence a robust property in almost periodic models? A well-behaved family: almost periodic Nicholson systems, Nonlinearity, 31 (2018), 388-413. doi: 10.1088/1361-6544/aa92e7.  Google Scholar [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [17] P. Poláčik and I. Tereščák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynamics Differential Equations, 5 (1993), 279-303. doi: 10.1007/BF01053163.  Google Scholar [18] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.  Google Scholar [19] R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations, 113 (1994), 17-67.  doi: 10.1006/jdeq.1994.1113.  Google Scholar [20] W. Shen and Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows, Mem. Amer. Math. Soc., 647, Amer. Math. Soc., Providence, 1998. doi: 10.1090/memo/0647.  Google Scholar [21] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, 1995.  Google Scholar [22] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3.  Google Scholar [23] C. C. Travis and G. F. Webb, Existence, stability, and compactness in the $\alpha$-norm for partial functional differential equations, Trans. Amer. Math. Soc., 240 (1978), 129-143. doi: 10.2307/1998809.  Google Scholar [24] J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

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##### References:
 [1] R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.  Google Scholar [2] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., 1964.  Google Scholar [3] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, Berlin, Heidelberg, New York, 1981.  Google Scholar [4] R. Johnson, R. Obaya, S. Novo, C. Núñez and R. Fabbri, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control, Developments in Mathematics 36, Springer, Switzerland, 2016. doi: 10.1007/978-3-319-29025-6.  Google Scholar [5] O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moskow, 1967 (Russian). English transl.: Transl. Math. Monographs, AMS, Providence, 1968.  Google Scholar [6] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications Vol. 16, Birkhäuser, Basel, Boston, Berlin, 1995. doi: 10.1007/978-3-0348-9234-6.  Google Scholar [7] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc. 321 (1990), 1-44. doi: 10.2307/2001590.  Google Scholar [8] R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35.   Google Scholar [9] J. Mierczyński and W. Shen, Lyapunov exponents and asymptotic dynamics in random Kolmogorov models, J. Evol. Equ., 4 (2004), 371-390. doi: 10.1007/s00028-004-0160-0.  Google Scholar [10] S. Novo, C. Núñez, R. Obaya and A. M. Sanz, Skew-product semiflows for non-autonomous partial functional differential equations with delay, Discrete Continuous Dynam. Systems - A, 34 (2014), 4291-4321.  doi: 10.3934/dcds.2014.34.4291.  Google Scholar [11] S. Novo, R. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dynamics Differential Equations, 25 (2013), 1201-1231. doi: 10.1007/s10884-013-9337-y.  Google Scholar [12] S. Novo, R. Obaya and A. M. Sanz, Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows, Nonlinearity, 26 (2013), 2409-2440.  doi: 10.1088/0951-7715/26/9/2409.  Google Scholar [13] C. Núñez, R. Obaya and A. M. Sanz, Minimal sets in monotone and sublinear skew-product semiflows Ⅱ: Two-dimensional systems of differential equations, J. Differential Equations, 248 (2010), 1899-1925. doi: 10.1016/j.jde.2009.12.006.  Google Scholar [14] R. Obaya and A. M. Sanz, Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems, J. Differential Equations, 261 (2016), 4135-4163. doi: 10.1016/j.jde.2016.06.019.  Google Scholar [15] R. Obaya and A. M. Sanz, Is uniform persistence a robust property in almost periodic models? A well-behaved family: almost periodic Nicholson systems, Nonlinearity, 31 (2018), 388-413. doi: 10.1088/1361-6544/aa92e7.  Google Scholar [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [17] P. Poláčik and I. Tereščák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynamics Differential Equations, 5 (1993), 279-303. doi: 10.1007/BF01053163.  Google Scholar [18] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.  Google Scholar [19] R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations, 113 (1994), 17-67.  doi: 10.1006/jdeq.1994.1113.  Google Scholar [20] W. Shen and Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows, Mem. Amer. Math. Soc., 647, Amer. Math. Soc., Providence, 1998. doi: 10.1090/memo/0647.  Google Scholar [21] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, 1995.  Google Scholar [22] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3.  Google Scholar [23] C. C. Travis and G. F. Webb, Existence, stability, and compactness in the $\alpha$-norm for partial functional differential equations, Trans. Amer. Math. Soc., 240 (1978), 129-143. doi: 10.2307/1998809.  Google Scholar [24] J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar
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